## Classification of circle actions on $4$-manifolds

HTML articles powered by AMS MathViewer

- by Ronald Fintushel PDF
- Trans. Amer. Math. Soc.
**242**(1978), 377-390 Request permission

## Abstract:

This article studies locally smooth ${S^1}$-actions on closed oriented 4-manifolds in terms of the orbit space, orbit type data, and the characteristic class of the action which lies in ${H_1}({M^{\ast }},{S^{\ast }})$ where ${M^{\ast }}$ is the orbit space and ${S^{\ast }}$ is the image of a certain collection of singular orbits. It is proved that such actions are determined by their weighted orbit spaces and are in 1-1 correspondence with “legally-weighted” 3-manifolds. The information contained in the weighted orbit space is used to give a presentation of the fundamental group of the 4-manifold, and in certain cases the quadratic form is computed.## References

- Glen E. Bredon,
*Introduction to compact transformation groups*, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR**0413144** - Philip T. Church and Klaus Lamotke,
*Almost free actions on manifolds*, Bull. Austral. Math. Soc.**10**(1974), 177–196. MR**370632**, DOI 10.1017/S000497270004082X - Ronald Fintushel,
*Locally smooth circle actions on homotopy $4$-spheres*, Duke Math. J.**43**(1976), no. 1, 63–70. MR**394716** - Ronald Fintushel,
*Circle actions on simply connected $4$-manifolds*, Trans. Amer. Math. Soc.**230**(1977), 147–171. MR**458456**, DOI 10.1090/S0002-9947-1977-0458456-6 - Wu-yi Hsiang,
*On the unknottedness of the fixed point set of differentiable circle group actions on spheres—P. A. Smith conjecture*, Bull. Amer. Math. Soc.**70**(1964), 678–680. MR**169238**, DOI 10.1090/S0002-9904-1964-11158-7 - Peter Orlik,
*Seifert manifolds*, Lecture Notes in Mathematics, Vol. 291, Springer-Verlag, Berlin-New York, 1972. MR**0426001**, DOI 10.1007/BFb0060329 - Peter Orlik and Frank Raymond,
*Actions of $\textrm {SO}(2)$ on 3-manifolds*, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 297–318. MR**0263112** - Frank Raymond,
*Classification of the actions of the circle on $3$-manifolds*, Trans. Amer. Math. Soc.**131**(1968), 51–78. MR**219086**, DOI 10.1090/S0002-9947-1968-0219086-9 - P. Orlik and P. Wagreich,
*Algebraic surfaces with $k^*$-action*, Acta Math.**138**(1977), no. 1-2, 43–81. MR**460342**, DOI 10.1007/BF02392313 - Peter Sie Pao,
*Nonlinear circle actions on the $4$-sphere and twisting spun knots*, Topology**17**(1978), no. 3, 291–296. MR**508892**, DOI 10.1016/0040-9383(78)90033-2 - Frank Raymond,
*Classification of the actions of the circle on $3$-manifolds*, Trans. Amer. Math. Soc.**131**(1968), 51–78. MR**219086**, DOI 10.1090/S0002-9947-1968-0219086-9 - H. Seifert,
*Topologie Dreidimensionaler Gefaserter Räume*, Acta Math.**60**(1933), no. 1, 147–238 (German). MR**1555366**, DOI 10.1007/BF02398271

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**242**(1978), 377-390 - MSC: Primary 57S15
- DOI: https://doi.org/10.1090/S0002-9947-1978-0496815-7
- MathSciNet review: 496815